A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Is this set of random variables a Hilbert space?

Consider a sequence of i.i.d. random variables $\left\{ {{\varepsilon _t}} \right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0$ and $E\left( {\varepsilon _t^2} \right) = {\sigma ...
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0answers
10 views

A simple implication of an approximation theorem by Komlós, Major and Tusnády

I have been reading through the PhD thesis of Professor Aue on change point analysis based on invariance principles. There's a particular argument I have not been able to follow: Let ...
0
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1answer
27 views

sufficient conditions for a stochastic process to be wide sense stationary

From the page Stationary process, I have the following definition: WSS random processes only require that 1st moment and autocovariance do not vary with respect to time and from the page ...
3
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0answers
29 views

Weak convergence of a sequence of stationary distributions to another stationary distribution

Let $\{X_n(t) \in \mathbb{Z}^+\}$ for each $t \in (0,1)$ denote a discrete time Markov chain (with time index $n$ and parameterized by $t$). For each $t$, the Markov chain $\{X_n(t)\}$ has a unique ...
4
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1answer
24 views

Largest jumps of a spectrally positive $\alpha$-stable process

Let $X(.)$ be a (strictly) $\alpha$-stable process (with $\alpha \in (1,2)$). Assume also that $X(.)$ is spectrally positive (its Lévy measure is concentrated in $[0,+\infty)$). I am looking for a ...
2
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3answers
46 views

compute temporal average of $\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)$

assuming that $\Phi$ is uniformly distributed over $(0,2\pi)$ compute: $$E[\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)]$$ I have solved the problem as continues: $$\begin{align} ...
0
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1answer
11 views

Hitting times of a biased continuous time random walk

Let $X_{s \geq 0}$ be a continuous time random walk on $\mathbb{Z}$, i.e. waiting times between jumps are exponentially distributed with mean one. The random walk is biased: $\mathbb{P}(X_s\text{ ...
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1answer
35 views

Why this process is nonergodic?

I am studying a tutorial on stochastic processes and there's an example in it which I don't understand anything of it. First of all there is this criterion for a mean-ergodic random process: For ...
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0answers
9 views

Continuity of the Loewner flow (SLE theory).

In the SLE paper "Basic Properties of the SLE" from Rohde and Schramm, it is mentioned on page 898 that the map $$(y,t)\mapsto g_t^{-1}(iy+\xi(t))$$ is clearly continuous on $y>0,t>0$, where ...
2
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0answers
32 views

Covariance of 1-D random process is $n\times n$!!!!

I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows: General Moving Average random process given as $X[n]=\frac{(U[n]+U[n-1])}{2}$ where $E[U[n]]=\mu$ ...
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0answers
21 views

Transition density of an AR(1) process?

If we have an AR(1) process, i.e: $X_{t+1} = \alpha X_t + e_{t+1}$ with $X_0=0$ then what is its Markov Chain transition density? We know that for a Markov chain, the following holds: $P(X_{t+1}\leq ...
2
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1answer
30 views

Is tossing a die in 10 consequent days an ergodic process?

IT maybe an elementary question but I'm totally new to the concept. In Wikipedia, ergodicity is defined as follows: In statistics, the term describes a random process for which the time ...
0
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1answer
19 views

what's the difference between variable and process from a statistical point of view?

I'm reading a tutorial stochastic process: ergodicity and temporal averages and I'm totally confused. It is said that: Suppose an IID random process whose marginal PDF is Gaussian with mean ...
1
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1answer
15 views

Discrete-time Markov Chains

I am having trouble understanding this proof from Markov Chains by Norris (1997) How do we get the equality $P_j(X_n=j \text{ for infinitely many } n ) =P_j(X_n=j \text { for some } n \ge m+1)$ ?
2
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1answer
27 views

Absolute continuity counterexample of a stochastic process

This example is from Stochastic Modelling and Applied Probability by Sören Asmussen (2010) p.358. The setup is the following: Let $\{Z_{t}\}$ be stochastic process on a Skorokhod space $D$ and a ...
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0answers
27 views

A Feynman-Kac style derivation of a survival probability of a Compound Poisson process

Let $$R_t = u + \beta t - \sum^{N_t}_{i=1}U_i$$where $u\geq 0$, $\beta > 0$, $N_t$ is a Poisson counting process with intensity $\lambda$ and $U_i$ are jumps having a probability density $\nu(y)$ ...
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13 views

Is the space of all adapted processes with Càdlàg paths a Banach space?

Consider first the definition of a stochastic integral for simple predictable processes. $$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$ The ...
3
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0answers
55 views

A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...
1
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1answer
37 views

Existence stochastic integral

I am trying to understand the prove of the existence of the stochastic integral for a local martinglale null at $0$ and continuous, $M\in \mathcal{M}^c_{0,\text{loc}}$, a predictable process $H\in ...
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1answer
15 views

Relationship between minimizing a conditional variance and a covariance

We are working with discrete-time stochastic processes. Let $v_k$ be a $\mathcal F_k$-predictable process, and let $X_k, \eta_k$ be $\mathcal F_k$-adapted processes. Define $V_k = v_kX_k+\eta_k$ and ...
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1answer
58 views

Show that $\mathbb{P}(\tau_{0}>T)\approx\frac{1}{\sqrt{T}}$ where $\{ B(t) : t\geq 0\}$ is a linear brownian motion started at $B(0)=1$ [on hold]

I'd appreciate if someone could provide me with a solution for the following problem: Let $\left\{ B\left(t\right)\thinspace|\thinspace t\geq0\right\}$ be a linear brownian motion started at ...
4
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2answers
64 views

How to take into account uncertainty on number of events

Suppose I generate a set of events $X_{i}$ for $i = 1,2 \dots N$ and suppose every event is either a success or a failure, ie. $X_{i} = 0, 1$. If $N$ is fixed, the MLE for the probability of success ...
4
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1answer
74 views
+50

Normalized hit times of a simple RW converge in distribution to hit times of standard Brownian Motion

I would appreciate some hints or guidance towards solving the following exercise: Let $\left\{ S\left(j\right)\thinspace:\thinspace j=0,1,...\right\}$ be a simple random walk on the integers ...
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0answers
8 views

How to simulate Permanental Point Process

I have simulated a determinantal point process in a square grid using Gaussian Kernel. The Gaussain matrix is decomposed into its eigenvectors and eigenvalues. In core implementation, the elementary ...
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0answers
26 views

Fundamental theorem for Malliavin derivative and Lebesgue integral

I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let ...
3
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5answers
126 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
1
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1answer
18 views

What is the probability of no events in a Markov-modulated Poisson process?

Suppose I have a two-state continuous-time Markov chain $M$ with rate matrix $Q$. $$ Q = \begin{bmatrix} -q_{01} & q_{01} \\ q_{10} & -q_{10} \end{bmatrix} $$ Now consider a Poisson process ...
4
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1answer
32 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Probability: Theory and Examples (4e), Theorem 8.4.1, page 317), there's a step which I'm a little shaky on. Basically, this proof ...
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0answers
10 views

Gaussian Process with explicit basis functions

I am considering the Gaussian process with explicit basis functions as discussed in the book (section 2.7): http://www.gaussianprocess.org/gpml/chapters/RW2.pdf Has anyone tried to derive formulas ...
1
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1answer
16 views

Martingale and independent increment

I know that in $L^2$ martingale a have independent increments. In particular that $\mathbb{E}[(X_m-X_n)^2]=\mathbb{E}[X^2_m-X^2_n]$ if X is a martingale. Does this extend also for general $p\geq 1$ in ...
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0answers
19 views

Simple Stratonovich product for physical system

I was reading a physical textbook and they used the "Stratonovich product" referred to $v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1$. I think this product is from the Stochastic process, thus ...
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0answers
14 views

Distribution of hitting time for two border brownian motion

I'm trying to find the distribution of hitting times for two border brownian motion with respect to both the hitting time AND which border is hit. Is this well defined? This is assuming $W_0=0$ with ...
1
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0answers
15 views

Smith's Key Renewal Theorem for Renewal Function

Consider a renewal process $(N_t)_{t \geq 0}$ and its renewal function $M(t):=\mathbb{E}[N_t]$ with interarrival distribution function $F$. One can show that $M$ satisfies the $(F,F)$-renewal ...
0
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1answer
73 views

Analytic solution to stochastic differential equations

I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ...
2
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0answers
20 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...
7
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1answer
143 views

When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$?

Suppose $\{X_1, X_2, X_3, \ldots\}$ is an infinite sequence of random variables such that $E[X_i]=0$ for all $i$, and $E[X_iX_j]=0$ whenever $i \neq j$. Further suppose the variances $\sigma_i^2 = ...
0
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1answer
17 views

Change from stochastic exponential to exponential of Lévy process - Applebaum

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ...
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0answers
13 views

Formula for running-time complexity

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$ Now it ...
2
votes
1answer
37 views

Independent increments of a Poisson process

In the following $\{X_t\}$ is a Poisson process. Assume that I've proved that $P(X_s=i,X_t-X_s=k)=P(X_s=i)P(X_t-X_s=k)$ so that the two events are independent, does it follow that ...
2
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1answer
27 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s ...
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0answers
26 views

Pure death processes

If $P_n (t)=\Pr (N (t)=n)$ and $N (0)=a$, how can I show that in a pure death process $$P_{(a-1)}(t)=a (e^{\mu t }-1)e^{-a \mu t}.$$ I showed that $P_a(t)=e^{-a \mu t}$. In fact I want to show ...
0
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1answer
28 views

Probability of time between two events in a poisson process

Suppose people arrive at a certain place according to a poisson process with rate 10 per day. 1) What is the expected time until the arrival of 100 person. 2) What is the probability that ...
3
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1answer
64 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
0
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1answer
18 views

Closed communicating class

Let $P_{ij}$ a transition matrix, a class $C$ is closed if given two different states $i$ and $j$ $$i\in C, i\rightarrow j\Rightarrow j\in C$$ If a Markov Chain is irreducible the transition matrix ...
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2answers
62 views

Probability in a fixed die

I have that transition matrix is ...
0
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1answer
44 views

Why is $f(X_t)-\int_0^t Af(X_s) \, ds$ a martingale for a Markov process $(X_t)_{t \geq 0}$?

I think if $A$ is the usual generator for the Markov process $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbb{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
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20 views

differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)} w.r.t$ t?

Here $\phi(u,t)=E\{e^{iut}\} $ is a characteristic function, $x_t$ is Gaussian. Differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)}$ w.r.t t the result is $\phi_u=\phi[i\hat{x}_t-P_tu]$
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vote
1answer
48 views

Understanding the Markov property of Brownian motion

I'm trying to understand the Markov property for Brownian motions in full generality. The textbook I'm following states it like this: Recall that we have a family of measures $P_x, x \in ...
0
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0answers
36 views

expected value of expected value

I want to quantify the error of phase noise in terms of its normalized mean squared error. I define the error measure as (x is the error free function, y the distorted): $$ \rm NMSE = \frac{\int ...
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0answers
78 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...